Problem solved, sort of. The mixture was too rich. This leads me to another question, though. Until now, I have been using some packing sponge inside the air box as an air filter. This is in addition to the metal screen thing that goes between the air box and the carburator. But these days, the moped will only run with that sponge taken out. Furthermore, I ordered a new air box, and it didn't have any filter in it (and there was no option of ordering such a filter). Now I'm wondering if this filter is actually supposed to be there at all, or if my dad just stuck it in when he was fixing the thing up. The moped is running great without the filter, but I'm worried about too much dust getting into the engine and messing things up.
So, if there is supposed to be a filter there, why isn't that sponge working (it breathes air ok)? If the carb is mixing too much gas in, and this is being corrected by having no air filter, then how can I decrease the amount of gas in the mixture?
To answer Duane: On level ground, my bike will go 30, no problem. It also says "Max speed 30mph" on the tire.